AI 中文总结
研究半线性波动方程中未知系数的恢复问题，核心方法是结合一阶线性化与容斥原理恒等式，利用波动方程几何光学解，还通过数值示例表明基于神经网络的反演算法可有效重建未知系数。
AI 中文摘要
本文研究在\(\mathbb{R}^{1 + n}\)（\(n \geq 2\)）中有界、开放且严格凸域上定义的半线性波动方程中未知系数的恢复问题。证明了在诺伊曼边界条件下半线性波动方程\(\square u + q u^m = 0\)中出现的未知系数\(q\)可从线性化的诺伊曼到狄利克雷（NtD）映射以赫尔德稳定性重建。方法结合一阶线性化与容斥原理（PIE）恒等式，在两种不同情况下使用波动方程的几何光学解。此外，数值示例表明在最小二乘法框架内使用基于神经网络的反演算法也可有效重建未知系数。
英文摘要
This paper investigates recovery of an unknown coefficient in a semilinear wave equation defined on a bounded, open, and strictly convex domain in \(\mathbb{R}^{1+n}\) with \(n \ge 2\). We demonstrate that the unknown coefficient \(q\) appearing in the semilinear wave equation \(\square u + q u^m = 0\) with Neumann boundary conditions can be reconstructed with Hölder stability from the linearized Neumann-to-Dirichlet (NtD) map. Our approach combines first-order linearization with the Principle of Inclusion-Exclusion (PIE) identity, and employs geometric optics solutions for wave equations in two distinct regimes: the case \(m=2\) with \(q = q(x)\), and the case \(m \ge 3\) with \(q = q(t,x)\). Furthermore, numerical examples illustrate that the unknown coefficient can also be effectively reconstructed using a neural network-based inversion algorithm within the framework of the least squares method.
